A didactic example is provided, using a Monte Carlo method, of how differential item functioning (DIF) can be eliminated (and thus better understood) when the complete latent space is used. The main source of DIF is that the matching single criterion used in some DIF procedures, Mantel Haenszel or Simultaneous Item Bias (SIBTEST), does not account for all of the abilities used by examinees in both groups of interest. To resolve this problem, several researchers have tried to match on secondary...

Topics: ERIC Archive, Computer Simulation, Equations (Mathematics), Item Bias, Item Response Theory,...

If equivalent operations are performed on the left term and right term of an equation, a new equation results. This principle allows one to produce equations with a variable isolated on one side and its value(s) on the other. It also underlies problem-solving in situations where equations are not explicitly used, but the problem calls for recognizing that two quantities are equal in value and using that information to derive conclusions about values of unknown quantities. The present paper...

Topics: ERIC Archive, Algebra, Elementary Secondary Education, Equations (Mathematics), Mathematics...

Although the WebQuest has been around for several years, it appears that not many educators are aware of it, or know how to use it. Thus, it is important to not only expose developmental educators to the use of this strategy, but also to help them realize how they can easily implement it into their curriculum. The use of a Webquest integrates technology into the classroom, while utilizing a learner-centered activity and a constructivist approach to learning. When using this strategy, the...

Topics: ERIC Archive, Technology Integration, Web Based Instruction, Educational Technology, Developmental...

A picture is worth more than a thousand words--in mathematics too. Many students fail in learning mathematics because, in some cases, teachers do not offer the necessary visualization. Nowadays technology overcomes this problem: computer aided instruction is one of the most efficients methods in teaching mathematics. In this article we try to discuss some methods of visualizing complex roots, with examples. The first method presented is the auxiliary polynomial method, followed by the the...

Topics: ERIC Archive, Mathematics Instruction, Computer Assisted Instruction, Visualization, Teaching...

This paper studies the modes of thought that occur during the act of solving problems in mathematics. It examines the two main instantiations of mathematical knowledge, the conceptual and the structural, and their role in the act. It claims that awareness of mathematical structure is the lever that educes mathematical knowledge existing in the mind in response to a problem-solving activity. For didactical purposes it proposes the consideration of mathematical techniques to facilitate the...

Topics: ERIC Archive, Cognitive Processes, Cognitive Style, Concept Formation, Curriculum Design,...

Differential item functioning (DIF) analysis is a key component in the evaluation of the fairness and validity of educational tests. The goal of this project was to review the status of ETS DIF analysis procedures, focusing on three aspects: (a) the nature and stringency of the statistical rules used to flag items, (b) the minimum sample size requirements that are currently in place for DIF analysis, and (c) the efficacy of criterion refinement. The main findings of the review are as follows:...

Topics: ERIC Archive, Test Bias, Sample Size, Bayesian Statistics, Evaluation Methods, Educational Testing,...

I describe a method of evaluating the integral of 1/r[superscript 2] with respect to r that uses only algebra and the concept of area underneath a curve, and which does not formally employ any calculus. This is useful for algebra-based introductory physics classes (where the use of calculus is forbidden) to derive the work done by the force of one point charge or mass on another. I extend the method to evaluate any integral of the form ?r[superscript n] dr for any integer n ? -1.

Topics: ERIC Archive, Science Instruction, Physics, Algebra, Scientific Concepts, Teaching Methods,...

Many bandpass signals can be sampled at rates lower than the Nyquist rate, allowing significant practical advantages. Illustrating this phenomenon after discussing (and proving) Shannon's sampling theorem provides a valuable opportunity for an instructor to reinforce the principle that innovation is possible when students strive to have a complete understanding of physical processes and mathematical models.

Topics: ERIC Archive, Engineering Education, Engineering, Acoustics, Mathematical Models, Equations...

The purpose of this article is to determine the levels of understanding for solving Stoichiometry problems from the aspect of macroscopic, microscopic and symbolic representations of high, average and low achieving students after infusion of metacognitive skills. Nine form four students aged sixteen years old from a secondary school in Kuala Lumpur, Malaysia were involved in this research. Data were collected through thinking aloud sessions and documents of students' worksheets. Findings showed...

Topics: ERIC Archive, Foreign Countries, Secondary School Students, Knowledge Level, Statistical Analysis,...

The paper discusses some interesting, mainly philosophical paradigms of the modeling and control areas, which are still partly unsolved and/or only partially studied. First the possible introduction of a prejudice free control--similar to the term for the modeling introduced by Rudi Kalman--is investigated. Next the real constraints in real control systems are discussed. It seems that these are the amplitude limit for the actuators in practical systems. Then the application of the...

Topics: ERIC Archive, Educational Philosophy, Scientific Research, Scientific Attitudes, Scientists,...

Making connections between various representations is important in mathematics. In this article, the authors discuss the numeric, algebraic, and graphical representations of sums of absolute values of linear functions. The initial explanations are accessible to all students who have experience graphing and who understand that absolute value simply means "distance from". (Contains 14 figures and 6 tables.)

Topics: ERIC Archive, Mathematics Instruction, Mathematical Concepts, Algebra, Secondary School...

A method was developed for identifying students who may be at high risk for failing the State Board Licensing Examination (SBE) for registered nurses. The subjects used in developing prediction equations included 50 students who graduated from the nursing program at Saint Joseph's College (SJC) in North Windham (Maine) during the years 1983-84. The validation group included 105 students who graduated from the nursing program at SJC during the years 1985-88. Subjects were predominately single,...

Topics: ERIC Archive, Certification, College Students, Equations (Mathematics), Failure, Grade Point...

We present a method to automatically detect collaborative patterns of student and tutor dialogue moves. The method identifies significant two-step excitatory transitions between dialogue moves, integrates the transitions into a directed graph representation, and generates and tests data-driven hypotheses from the directed graph. The method was applied to a large corpus of student-tutor dialogue moves from expert tutoring sessions. An examination of the subset of the corpus consisting of tutor...

Topics: ERIC Archive, Tutors, Interpersonal Communication, Discourse Analysis, Intelligent Tutoring...

This paper describes the graded response model. The graded response model represents a family of mathematical models that deal with ordered polytomous categories, such as: (1) letter grading; (2) an attitude survey with "strongly disagree, disagree, agree, and strongly agree" choices; (3) partial credit given in accord with an individual's degree of attainment; and (4) computerized cognitive diagnosis. Equations are given for the general model and for homogeneous and heterogeneous...

Topics: ERIC Archive, Equations (Mathematics), Estimation (Mathematics), Mathematical Models, Selection,...

Padula, J. (2005) described how mathematics teachers can channel the passion of students interested in electronic music by teaching the mathematics that has been used to create it. She hypothesised that the study and enjoyment of music may help the study of mathematics since both mathematics and music are symbol systems and pattern (and its recognition) are important to both. In this article, the author discusses how to teach fractal geometry with music.

Topics: ERIC Archive, Music, Geometric Concepts, Mathematics Teachers, Geometry, Teaching Methods,...

Participants estimated the imagined elongation of a spring while they were imagining that a load was stretching the spring. This elongation turned out to be a multiplicative function of spring length and load weight--a cognitive law analogous to Hooke's law of elasticity. Participants also estimated the total imagined elongation of springs joined either in series or in parallel. This total elongation was longer for serial than for parallel springs, and increased proportionally to the number of...

Topics: ERIC Archive, Mechanics (Physics), Mathematical Concepts, Computation, Scientific Principles,...

Usually a student learns to solve a system of linear equations in two ways: "substitution" and "elimination." While the two methods will of course lead to the same answer they are considered different because the thinking process is different. In this paper the author solves a system in these two ways to demonstrate the similarity in the computation. She then sees that changing the point of view leads to a "simpler" way to solve a system of equations. This leads...

Topics: ERIC Archive, Equations (Mathematics), Matrices, Mathematics, Mathematics Instruction, Problem...

The adequacy of several approaches to estimation of proficiency distributions for the Trial State Assessment (TSA) in eighth grade mathematics of the National Assessment of Educational Progress was examined. These approaches are more restrictive than the estimation procedures originally used, with the same kind of plausible-values approach that has been implemented since 1984 with the national assessment. The proposed approaches are computationally less burdensome and could provide improved...

Topics: ERIC Archive, Academic Achievement, Comparative Analysis, Competence, Equations (Mathematics),...

In this paper are described some features of the intensive use of math software, primarily DERIVE, in the context of modeling in an introductory university course in differential equations. Different aspects are detailed: changes in the curriculum that included not only course contents, but also the sequence of introduction to various topics and methodologies. For example, the use of software at the beginning of the course followed a black box strategy, in order to emphasize mathematical...

Topics: ERIC Archive, Calculus, Computer Uses in Education, Differential Equations, Higher Education,...

Teachers use action research in order to improve their teaching and student learning. This action research will analyze students' algebraic reasoning in finding values of variables in systems of equations pictorially and algebraically. This research will look at students solving linear systems of equations without knowing the algebraic algorithms. It will concentrate on the students' constructive learning abilities to resolve and create methods in solving linear systems of equations. The study...

Topics: ERIC Archive, Action Research, Grade 8, Equations (Mathematics), Algebra, Mathematics Instruction,...

Early algebraic thinking in a primary context is not about introducing formal algebraic concepts into the classroom but involves reconsidering how one thinks about arithmetic. Early algebraic thinking assists young students to engage effectively with arithmetic in ways that support engagement with arithmetic structure rather than arithmetic as a tool for computation. The distinction between arithmetic thinking and algebraic thinking in the early years' context is best defined as: arithmetic...

Topics: ERIC Archive, Equations (Mathematics), Arithmetic, Algebra, Computation, Mathematical Formulas,...

The phenomenon of simultaneous differential item functioning (DIF) amplification and cancellation and the role of the SIBTEST computer program in detecting it were studied. A variety of simulated test data was generated for this purpose. In addition, the following real test data were used: (1) American College Testing program data for 2,115 males and 2,885 females in mathematics; (2) National Assessment of Educational Progress (NAEP) history test data for 1,225 males and 1,215 females; and (3)...

Topics: ERIC Archive, Black Students, Comparative Testing, Computer Simulation, Computer Software,...

This article is about a very small subset of the positive integers. The positive integer N is said to be "perfect" if it is the sum of all its divisors, including 1, but less that N itself. For example, N = 6 is perfect, because the (relevant) divisors are 1, 2 and 3, and 6 = 1 + 2 + 3. On the other hand, N = 12 has divisors 1, 2, 3, 4 and 6, but since 1 + 2 + 3 + 4 + 6 = 16, rather than 12, 12 is not a perfect number. Ways and exercises for finding perfect numbers are presented....

Topics: ERIC Archive, Number Concepts, Arithmetic, Equations (Mathematics), Mathematics Instruction, Scott,...

Most of the multivariate statistical techniques rely on the assumption of multivariate normality. The effects of non-normality on multivariate tests are assumed to be negligible when variance-covariance matrices and sample sizes are equal. Therefore, in practice, investigators do not usually attempt to remove non-normality. In this simulation study, the effects of non-normality on skewed multivariate data in terms of power were examined by manipulating the factors such as distribution, sample...

Topics: ERIC Archive, Computer Simulation, Equations (Mathematics), Mathematical Models, Matrices,...

Logarithms continue to play an important role in mathematics (most significantly in calculus), science, and engineering. It is therefore important for students to understand logarithms as real numbers as well as the characteristics of logarithmic functions. Exploration of challenges in understanding logarithms as real numbers and logarithmic functions as well as their graphs provides insight that can be used as the basis for instruction. This paper discusses and shares evidence of students'...

Topics: ERIC Archive, Numbers, Mathematics, Mathematics Instruction, Calculators, Mathematical Concepts,...

When a shock wave interacts with a contact discontinuity, there may appear a reflected rarefaction wave, a deflected contact discontinuity and a refracted supersonic shock. The numerical simulation of shock wave refraction at a plane contact discontinuity separating gases with different densities is performed. Euler equations describing inviscid compressible flow were discretized using the finite volume method on unstructured meshes and WENO schemes. Time integration was performed using a...

Topics: ERIC Archive, Simulation, Equations (Mathematics), Scientific Concepts, Science Instruction,...

Studies have shown that students' understanding is typically analytic and not visual. Two possible reasons for this are when the analytic mode, instead of the graphic mode, is most frequently used in instruction or, when students or teachers hold the belief that mathematics consists simply of skillful manipulation of symbols and numbers. The National Council of Teachers of Mathematics (NCTM) states that: "Different representations support different ways of thinking about and manipulating...

Topics: ERIC Archive, Symbols (Mathematics), Mathematics Teachers, Geometric Concepts, Arithmetic,...

Modeling the population in survey sampling problems continues to be controversial. An important reason is that the likelihood principle makes it somewhat necessary to model the population. Estimating the population total in two-stage survey sampling is considered, making use of a "superpopulation" model. The problem is then really one of predicting the unobserved part of the total. The concept of predictive likelihood is studied. A non-Bayesian likelihood approach to prediction...

Topics: ERIC Archive, Equations (Mathematics), Mathematical Models, Maximum Likelihood Statistics,...

In this study, the authors wanted to examine whether success on items testing basic equivalence knowledge, such as the meaning of the equal sign and ability to solve problems such as 3 + 5 = 4 + _, predicted success on items testing more advanced algebraic thinking (i.e. principles of equality and solving equations that use letter variables). This investigation is a follow up study to their initial efforts to design an instrument to measure children's understanding of equivalence...

Topics: ERIC Archive, Algebra, Replication (Evaluation), Item Response Theory, Equations (Mathematics),...

Twelve linear algebra students were interviewed about the concept of a Solution of a System of Equations. The interviews were analyzed using APOS tools, in particular the ideas of Action, Process, Object and Schema, and Genetic Decomposition. The analysis of the interviews revealed several misconceptions of Solution. The analysis also revealed shortcomings of the questionnaire that was used in the interviews: It did not permit making a distinction between lack of knowledge and partial...

Topics: ERIC Archive, Research Tools, Misconceptions, Algebra, Problem Solving, Equations (Mathematics),...

Some of the conceptual qualitative ideas needed to test nonlinear models empirically and to modify them are described. Relationships among these ideas and computer applications are also examined to elucidate the general process of nonlinear modeling. Two examples are presented along with a discussion of bifurcation, catastrophe, and maximum likelihood estimate methods. The first example concerns administrators' responses to innovation and uses a verbal description of events. The model is...

Topics: ERIC Archive, College Administration, Equations (Mathematics), Estimation (Mathematics), Higher...

With a male, adult subject, this study measured the punch force and kick force of the Japanese martial art Nihon-Kempo, the movements of the dominant (right) arm, leg, and the movements of the legs using timed synchronization. Using an electromyogram (EMG), the relationship between the variation of punch force and the variation of the electrical potential of the muscles were examined. It was observed that when the punch force was at maximum, the electrical potential of the muscles was not at...

Topics: ERIC Archive, Motion, Equations (Mathematics), Perceptual Motor Coordination, Physical Activity...

This study aimed to determine students' metacognitive failure in Mathematics Education Program of FKIP in Jambi University investigated based on assimilation and accommodation Mathematical framework. There were 35 students, five students did not answer the question, three students completed the questions correctly and 27 students tried to solve problems but unfortunately made the same mistake. Out of 27 students involved in the study, two students were taken as the research subjects. The...

Topics: ERIC Archive, Metacognition, Mathematics Education, Problem Solving, Qualitative Research,...

Perhaps a business colleague threw out a challenge. The year: around 1200. The place: Pisa. The challenge: Calculate how many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on. The question and its solution found its way into the book "Liber abaci" by Leonardo of Pisa (known as Fibonacci), completed in 1202. It gives rise to the Fibonacci sequence. A colleague of the...

Topics: ERIC Archive, Number Concepts, Mathematics Instruction, Secondary School Mathematics, Validity,...

Various theoretical equations have been proposed to predict response rate as a function of the rate of reinforcement. If both the rate and probability of reinforcement are considered, a simple identity, defining equation, or "law" holds. This identity places algebraic constraints on the allowable forms of our mathematical models and can help identify the referents for certain empirical or theoretical coefficients. This identity can be applied to both single and compound schedules of...

Topics: ERIC Archive, Behavioral Science Research, Economics, Responses, Reinforcement, Probability,...

In this article, the author states that architects, musicians and other thoughtful people have, since the time of Pythagoras, been fascinated by various harmonious proportions. One, is the visual harmony attributed to Euclid, called "the golden section". He explores this concept in geometries of one, two and three dimensions. He added, that in order to fully appreciate the three-dimensional proportions, model bases and capitals might be constructed from sheet-board and be displayed.

Topics: ERIC Archive, Geometric Concepts, Geometry, Equations (Mathematics), Visual Perception, Stuart,...

A predictive adaptive testing (PAT) strategy was developed based on statistical predictive analysis, and its feasibility was studied by comparing PAT performance to those of the Flexilevel, Bayesian modal, and expected a posteriori (EAP) strategies in a simulated environment. The proposed adaptive test is based on the idea of using item difficulty and past information (observed data) about an examinee to acquire the probability of answering further items correctly. Development of the PAT model...

Topics: ERIC Archive, Adaptive Testing, Bayesian Statistics, Comparative Analysis, Computer Assisted...

Research efforts have established that the Mantel-Haenszel procedure (MHP) is an effective method for detecting the presence of test items exhibiting differential item functioning (DIF). While the MHP has been advocated for situations where item response theory based methods may not be usable, recent findings have suggested that the performance of the MHP and Logit needs to be examined in detail. The present research examined the impact of manipulating sample size, the ratio of focal to...

Topics: ERIC Archive, College Entrance Examinations, Comparative Analysis, Control Groups, Equations...

Concurrent schedules were used to establish 6 hens' preferences for three foods. The resulting biases suggested wheat was preferred over honey-puffed and puffed wheat, and puffed wheat was the least preferred food. The hens then responded under fixed-ratio schedules for each food in 40-min (excluding reinforcer time) sessions, with the response requirement doubling each session until no reinforcers were received. At the smaller ratios, the less preferred the food, the faster the hens' overall...

Topics: ERIC Archive, Reinforcement, Food, Animals, Behavioral Science Research, Reaction Time, Responses,...

This paper focuses on three aspects related to the conceptualization and application of canonical correlation analysis as a dominant statistical model: (1) partial canonical correlation analysis and its application in statistical testing; (2) the relation between canonical correlation analysis and discriminant analysis; and (3) the relation between canonical correlation analysis and chi-square contingency table analysis. The paper shows that canonical correlation analysis can be conceptualized...

Topics: ERIC Archive, Chi Square, Classification, Computer Oriented Programs, Correlation, Discriminant...

This paper compares the way lessons on systems of linear equations unfold in a classroom in the Negev region of Israel with the way they unfold in a Shanghai and Hong Kong classroom. Lessons are viewed as temporal entities describable not only by the nexus of topics they contain but also by how they flow in time. In this light, the lessons in the classroom studied by the authors contrasts strongly with the Shanghai and Hong Kong classroom, the former having a turbulent flow and latter a smooth...

Topics: ERIC Archive, Foreign Countries, Grade 8, Cultural Differences, Algebra, Equations (Mathematics),...

The Purposeful Algebraic Activity Project is a longitudinal study of the development of pupils' algebraic activity in the early years of their secondary schooling. Here, we report on our empirical findings from the initial semi-structured interviews. We analyse the responses of three pairs of 12-year-old pupils to a range of algebra questions. In our analysis, we identify broad similarities in the "answers" pupils gave to transformational questions and quite significant differences in...

Topics: ERIC Archive, Mathematical Concepts, Interviews, Algebra, Competence, Mathematics Instruction,...

Since 1985 several procedures for computerized test construction using linear programing techniques have been described in the literature. To apply these procedures successfully, suitable item banks are needed. The problem of designing item banks based on item response theory (IRT) is addressed. A procedure is presented that determines whether an existing item bank meets the test construction requirements. If not, the method indicates which items have to be added to the banks so that it will...

Topics: ERIC Archive, Equations (Mathematics), Foreign Countries, Item Banks, Item Response Theory, Linear...

This paper describes how a simple application of de Moivre's theorem may be used to not only find the roots of a quadratic equation with real or generally complex coefficients but also to pinpoint their location in the Argand plane. This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not make the full visual connection between the Cartesian plane and the Argand plane that Bardell's three dimensional surfaces illustrated so well.

Topics: ERIC Archive, Mathematics Instruction, Mathematical Logic, Equations (Mathematics), Problem...

The article offers the educational information quantization method for improving content quality in Learning Management Systems. The paper considers questions concerning analysis of quality of quantized presentation of educational information, based on quantitative text parameters: average frequencies of parts of speech, used in the text; formal text readability indexes; lexical and syntactic text variety factors. The process of obtaining quantitative parameter values is focused on use of the...

Topics: ERIC Archive, Integrated Learning Systems, Management Systems, Electronic Learning, Educational...

The indirect method for preparing the statement of cash flows, as described in many standard textbooks, involves an item-by-item approach, telling you to add to or subtract from the net income, the increases or decreases in the balance sheet items, such as accounts payable or accounts receivable. Many business students, especially at the undergraduate level, find these black-box-rules confusing. In recent years, several articles have appeared in the accounting literature, exploring the link...

Topics: ERIC Archive, Algebra, Business Administration Education, Accounting, Money Management, Financial...

For the last three years the author of this paper has been teaching mathematics to a group of students at Year 9 level who have rarely had success in mathematics, particularly in the area of algebra. Most of these students struggle with algebraic concepts and processes and generally have poor retention of any acquired knowledge. For the two years prior to this year the students in the class have struggled with the topic of solving linear equations, using incorrect operations to solve the...

Topics: ERIC Archive, Mathematics Instruction, Algebra, Equations (Mathematics), Grade 9, Computer Assisted...

As time has progressed, the role of applied mathematics has become increasingly important. Indeed there are now more students enrolled in applied mathematics courses in senior high schools and colleges than in pure mathematics. Such courses become more relevant both to the student and to future employers, if the same constants and equations that are used in industry are reflected in the student's texts. The appropriate formulae for these constants and equations are featured in this article.

Topics: ERIC Archive, Mathematics Instruction, Equations (Mathematics), Geometry, Trigonometry, Earth...

Following a pretest, 8 participants who were unfamiliar with algebraic and trigonometric functions received a brief presentation on the rectangular coordinate system. Next, they participated in a computer-interactive matching-to-sample procedure that trained formula-to-formula and formula-to-graph relations. Then, they were exposed to 40 novel formula-to-graph tests and 10 novel graph-to-formula tests. Seven of the 8 participants showed substantial improvement in identifying formula-to-graph...

Topics: ERIC Archive, Equations (Mathematics), Cues, Trigonometry, Algebra, Mathematical Formulas, Pretests...

Three important methods areas of multivariate analysis that are not always thought of in terms of latent variable constructs, but for which latent variable modeling can be used to great advantage, are discussed. These methods are: (1) random coefficients describing individual differences in growth; (2) unobserved variables corresponding to missing data; and (3) variance components describing data from cluster sampling. An educational achievement dataset of longitudinal observations on secondary...

Topics: ERIC Archive, Achievement Tests, Cluster Analysis, Equations (Mathematics), Individual Differences,...